One approach to condition discriminator \(D\)  on \({\bf {g}}\)  is using multilinear map, which is the outer product \({\bf {x}} \otimes {\bf {y}}\)  of two vectors and is superior than concatenation [1]}. However, it results in dimension explosion and the output dimension is \(m \times d \times 2m\)  in our application, which is impossible to be embedded. Following CDAN, we tackle it in a randomized approach. The multilinear map of two pairs of features and outputs can be approximated by
\(\left\langle {{\bf {f}} \otimes {\bf {g}},{\bf {f}}^{\prime } \otimes {\bf {g}}^{\prime }} \right\rangle \approx \left\langle {{Z_R}({\bf {f}},{\bf {g}}),{Z_R}({\bf {f}}^{\prime },{\bf {g}}^{\prime })} \right\rangle ,\) 
